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Here is my guess. To compare spaces with their notion of strict $\infty$-groupoids (in which everything is strict except inverses) Kapranov and Voevodsky use an intermediate category of Kan diagrammat …

In a project in progress with Matan Prasma and Joost Nuiten concerning the abstract cotangent complex formalism we compute the stabilization of the $\infty$-category $\infty\mathrm{Cat}_{/C}$ of $\inf …

I believe that this is a particular case of Lurie's "operadic left Kan extension". We may identify a monoidal $\infty$-category $\mathcal{C}$ with a coCartesian fibrations of $\infty$-operads $\mathca …

In Appendix A.3 of the book higher topos theory appears the notion of an excellent model category (see Definition A.3.2.16). The main feature of this notion is that when $\mathbf{S}$ is an excellent m …

It is now known that the $\infty$-category of $\infty$-operads as defined by Lurie is equivalent to the underlying $\infty$-category of the model category of simplicial operads, see http://arxiv.org/p …

In Appendix C of his book in progress Spectral Algebraic Geometry, Lurie defines the unseparated derived category $\check{{\cal D}}({\cal A})$ (see Definition C.5.8.2 loc.cit) associated to a Grothend …

One way to construct the duality functor ${\cal C} \to {\cal C^{\rm op}}$ is through the notion of a pairing of $\infty$-categories (see HA, Definition 5.2.1.5). In particular, in this case we're talk …

The (epi,mono) factorization system in Sets is part of a model structure on Sets whose weak equivalences are the epis, fibrations are monos and cofibrations are everything. This is a model for the hom …

I think there is a typo in Lemma 5.4.5.11: $K$ is supposed to be $\tau$-small and not $\kappa$-small. Note that if $\tau < \kappa$ and $K$ is $\kappa$-small but not $\tau$-small then the statement of …

Following up on the answer of Simon Henry, let us prove the following statement. For a pro-space $\hat{X} = \{X_i\}_{i \in I}$, we let $Spaces_{/\hat{X}}$ denote the $\infty$-topos defined as the (co …

When working with quasi-categories, it is often more convenient (and more compatible with existing machinery) not to work with actual strict diagrams of quasi-categories but rather with coCartesian fi …

Let us prove the analogous claim for weak 2-groups (this implies, in particular, the strict case). Let $\mathcal{C}$ be a monoidal category with unit $\mathbb{I} \in \mathcal{C}$. Claim: Suppose the …

Technically speaking the answer to your question is no, in the sense that the data of $(\alpha,\beta,\gamma,\delta)$ alone does not determine the filtered object $V_0 \subseteq V_1 \subseteq V_2$. How …

I think what Lurie might have meant when he wrote "It is easy to see that $St_{\phi}$ preserves cofibrations" in the proof of Theorem 2.2.1.2, is that it is easy to see it if you take into account the …

The problem with this definition is that the formula $\mathrm{colim}_{X \to Z \in \mathcal{W}} F(Z)$ does not, in general, depend functorially on $X$. For it to depend functorially on $X$ you need a w …